How to keyframe "Use Count" in particle system?
You have keyframed the checkbox that enables or disables 'Use Count', not the 'Count' values themselves, which are shown below the large yellow box in your screenshot.

However, trying to keyframe the actual 'Count' value displays an error message, which indicates that these values probably can't be animated.

I would suggest posting another question on this site asking specifically about how to get different amounts of particles to appear.